{"id":26686,"date":"2024-08-30T17:22:05","date_gmt":"2024-08-30T17:22:05","guid":{"rendered":"https:\/\/science-hub.click\/%E7%99%BA%E6%95%A3%28%E6%95%B0%E5%AD%A6%29%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC\/"},"modified":"2024-08-30T17:22:05","modified_gmt":"2024-08-30T17:22:05","slug":"%E7%99%BA%E6%95%A3%28%E6%95%B0%E5%AD%A6%29%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=26686","title":{"rendered":"\u767a\u6563 (\u6570\u5b66)\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac"},"content":{"rendered":"

\u30c0\u30a4\u30d0\u30fc\u30b8\u30a7\u30f3\u30b9\u6f14\u7b97\u5b50\u306f<\/strong>\u3001\u4e00\u6b21\u504f\u5fae\u5206\u3092\u4f34\u3046\u7dda\u5f62\u5fae\u5206\u6f14\u7b97\u5b50\u3067\u3042\u308a\u3001\u7279\u306b\u4fdd\u5b58\u5247\u3092\u8868\u3059\u305f\u3081\u306b\u7269\u7406\u5b66\u3067\u3088\u304f\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002\u3053\u308c\u306f\u30d9\u30af\u30c8\u30eb\u5834\u3092\u30b9\u30ab\u30e9\u30fc\u5834 (\u3064\u307e\u308a\u95a2\u6570) \u306b\u5909\u63db\u3057\u3001\u3088\u308a\u4e00\u822c\u7684\u306b\u306f\u6b21\u6570k<\/i><\/span>\u306e\u30c6\u30f3\u30bd\u30eb\u5834\u3092<\/a><\/span>\u6b21\u6570k<\/i> \u2212 1<\/span>\u306e\u5834\u306b\u5909\u63db\u3057\u307e\u3059\u3002<\/p>

\u30d9\u30af\u30c8\u30eb\u5834<\/a><\/span>\u306e\u767a\u6563<\/span><\/h2>

\u6b21\u5143<\/a><\/span>3<\/strong>\u304a\u3088\u3073 \u30c7\u30ab\u30eb\u30c8\u5ea7\u6a19<\/a><\/span>\u3067\u3001\u30d9\u30af\u30c8\u30eb\u5834\u306e\u767a\u6563\u3092\u5b9a\u7fa9\u3057\u307e\u3059\u3002

$$ {\\vec A} $$<\/div>\u95a2\u4fc2\u306b\u3088\u3063\u3066<\/p>
$$ {\\mathrm{div}\\vec A = \\frac{\\part{A_x}}{\\part{x}}+\\frac{\\part{A_y}}{\\part{y}}+\\frac{\\part{A_z}}{\\part{z}}} $$<\/div><\/dd><\/dl>

\u6b63\u5f0f\u306b\u306f\u3001\u767a\u6563\u6f14\u7b97\u5b50\u306f<\/a><\/span>\u30d9\u30af\u30c8\u30eb\u5834\u306b\u9069\u7528\u3055\u308c\u307e\u3059\u3002

$$ {\\vec A} $$<\/div>\u30d9\u30af\u30c8\u30ebnabla<\/a><\/span>\u306e\u30b9\u30ab\u30e9\u30fc\u7a4d\u3067\u3082\u3042\u308a\u307e\u3059
$$ {\\vec\\nabla} $$<\/div>\u30d9\u30af\u30c8\u30eb\u306b\u3088\u3063\u3066
$$ {\\vec A} $$<\/div> \u3002 <\/p>
$$ {\\vec\\nabla \\cdot \\vec A = \\mathrm{div}\\vec A = \\frac{\\part{A_x}}{\\part{x}}+\\frac{\\part{A_y}}{\\part{y}}+\\frac{\\part{A_z}}{\\part{z}}} $$<\/div><\/dd><\/dl>

\u3068\u8a00\u3046\u306e\u3082\u540c\u3058\u3053\u3068\u3067\u3059

$$ {\\mathrm{div}\\vec A = \\mathrm{Tr}(\\mathrm D\\vec A)} $$<\/div> \u3002<\/p>

\u3053\u3053\u3001\u30d5\u30a3\u30fc\u30eb\u30c9

$$ {\\vec A} $$<\/div>\u306e\u5fdc\u7528\u3068\u307f\u306a\u3055\u308c\u307e\u3059
$$ {\\R^3} $$<\/div>\u81ea\u5206\u81ea\u8eab\u306e\u4e2d\u3067\u3001\u305d\u3057\u3066
$$ {\\mathrm D\\vec A} $$<\/div>\u306f\u5fae\u5206\u3092\u6307\u5b9a\u3057\u3001\u305d\u3053\u304b\u3089\u30c8\u30ec\u30fc\u30b9\u3001\u3064\u307e\u308a\u30e4\u30b3\u30d3\u30a2\u30f3\u884c\u5217\u306e\u30c8\u30ec\u30fc\u30b9\u3092\u53d6\u5f97\u3057\u307e\u3059\u3002<\/p>

\u3053\u306e\u6700\u5f8c\u306e\u30d7\u30ec\u30bc\u30f3\u30c6\u30fc\u30b7\u30e7\u30f3\u306b\u306f\u3001\u30d9\u30fc\u30b9\u306e\u9078\u629e\u306b\u4f9d\u5b58\u3057\u306a\u3044\u3068\u3044\u3046\u5229\u70b9\u304c\u3042\u308a\u307e\u3059\u3002\u4e0a\u8a18\u306f\u3059\u3079\u3066<\/a><\/span>\u6b21\u306e\u5834\u5408\u306b\u3082\u6709\u52b9\u3067\u3059

$$ {\\R^n} $$<\/div><\/p>

\u4f8b<\/strong><\/p>

\u3082\u3057
$$ {\\textstyle\\vec A=\\sum_{i=1}^n A_i\\frac{\\partial}{\\partial x_i}} $$<\/div>\u306f\u7dda\u5f62\u5834\u3067\u3059\u3002\u3064\u307e\u308a\u3001
$$ {\\textstyle A_i=\\sum_{j=1}^n a_{ij}x_j} $$<\/div> \u3001\u305d\u306e\u767a\u6563\u306f\u4ee5\u4e0b\u306b\u7b49\u3057\u3044
$$ {\\textstyle\\sum_{i=1}^na_{ii}} $$<\/div> \u3002\u3053\u308c\u306f\u3001\u6570\u5024a<\/i> i<\/i> j<\/i><\/sub><\/span> (
$$ {1\\le i,j\\le n} $$<\/div> \uff09\u3002<\/dd><\/dl>

\u4e56\u96e2\u306e\u89e3\u91c8<\/strong><\/p>

\u767a\u6563\u306f\u6d41\u308c<\/a><\/span>\u306e\u89b3\u70b9\u304b\u3089\u89e3\u91c8\u3055\u308c\u307e\u3059\u3002 D<\/i><\/span>\u304c\u6b21\u306e\u30c9\u30e1\u30a4\u30f3\u306e\u5834\u5408

$$ {\\R^3} $$<\/div>\u30a8\u30c3\u30b8S<\/i><\/span> \u3001\u306e\u6d41\u308c
$$ {\\vec A} $$<\/div>\u30b9\u30c8\u30fc\u30af\u30b9\u306e\u5b9a\u7406<\/a><\/span>\u306b\u3088\u308c\u3070\u3001 S<\/i><\/span>\u3092\u901a\u308b\u5024\u306f\u767a\u6563\u306eD<\/i><\/span>\u306b\u308f\u305f\u308b\u7a4d\u5206<\/a><\/span>\u306b\u7b49\u3057\u3044\u3002<\/p>

\u95a2\u9023\u3059\u308b\u89e3\u91c8\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 \u03c6 t \u3092<\/i><\/sub><\/span>\u5834\u306e\u6d41\u308c\u3068\u3059\u308b

$$ {\\vec A} $$<\/div> (\u3064\u307e\u308a\u3001\u5fae\u5206<\/a><\/span>\u30b7\u30b9\u30c6\u30e0\u306e\u89e3\u306e\u6642\u9593t<\/i><\/span>\u3067\u306e\u5024
$$ {\\frac{\\mathrm du}{\\mathrm dt}=\\vec A\\bigl(u(t)\\bigr)} $$<\/div>\u3053\u308c\u306f0<\/span>\u3067\u306ex<\/i><\/span>\u306e\u4fa1\u5024\u304c\u3042\u308a\u307e\u3059) <\/p>
$$ {L_A(\\mathrm dx\\wedge\\mathrm dy\\wedge\\mathrm dz)= \\mathrm{div}\\vec A(\\mathrm dx\\wedge\\mathrm dy\\wedge\\mathrm dz)} $$<\/div><\/dd><\/dl>

(\u30ea\u30fc\u5fae\u5206<\/a><\/span>\u6f14\u7b97\u5b50\u3092L<\/i> A<\/i><\/sub><\/span>\u3067\u6307\u5b9a\u3057\u307e\u3059\u3002\u8a73\u7d30\u304a\u3088\u3073\u3088\u308a\u4e00\u822c\u7684\u306a\u30b9\u30c6\u30fc\u30c8\u30e1\u30f3\u30c8\u306b\u3064\u3044\u3066\u306f\u3001\u30e9\u30d7\u30e9\u30b9 \u30d9\u30eb\u30c8\u30e9\u30df\u6f14\u7b97\u5b50\u3092\u53c2\u7167\u3057\u3066\u304f\u3060\u3055\u3044)\u3002<\/p>

\u7279\u306b\u3001\u4ee5\u4e0b\u306e\u6d41\u308c\u306f\u3001

$$ {\\vec A_,} $$<\/div>\u4f53\u7a4d\u3092\u7bc0\u7d04\u3057\u307e\u3059\uff08\u3064\u307e\u308a\u3001
$$ {\\mathrm{vol}\\big(\\phi_t(D)\\big) =\\mathrm{vol}(D)} $$<\/div>\u4efb\u610f\u306e\u30c9\u30e1\u30a4\u30f3D<\/i><\/span>\u306b\u3064\u3044\u3066)\u3001\u3069\u3053\u3067\u3082\u767a\u6563\u304c\u30bc\u30ed\u3067\u3042\u308b\u5834\u5408\u306b\u9650\u308a\u307e\u3059\u3002\u767a\u6563\u304c\u6b63\u306e\u5834\u5408\u306f\u30dc\u30ea\u30e5\u30fc\u30e0\u304c\u5897\u52a0\u3057\u3001\u8ca0\u306e\u5834\u5408\u306f\u30dc\u30ea\u30e5\u30fc\u30e0\u304c\u6e1b\u5c11\u3057\u307e\u3059\u3002<\/p>

\u3088\u308a\u4e00\u822c\u7684\u306b\u306f\u3001\u767a\u6563\u306f\u70b9<\/a><\/span>\u306e\u5468\u308a\u306e<\/a><\/span>\u4f53\u7a4d (\u307e\u305f\u306f\u96fb\u8377) \u306e\u5fae\u5c0f\u306a\u5909\u5316<\/strong>\u3092<\/a><\/span>\u8aac\u660e\u3057\u3001\u6d41\u4f53\u529b\u5b66<\/a><\/span>\u306e\u65b9\u7a0b\u5f0f\u307e\u305f\u306f\u30de\u30af\u30b9\u30a6\u30a7\u30eb\u65b9\u7a0b\u5f0f<\/span>\u3078\u306e\u305d\u306e\u4ecb\u5165\u3092\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>

\n\"\u767a\u6563<\/figure>

\u30b3\u30e1\u30f3\u30c8\u4ed8\u304d\u30d5\u30a9\u30fc\u30e0<\/span><\/h2>
$$ {\\mathrm{div}(\\mathrm{rot}\\vec A) =0} $$<\/div><\/dd><\/dl>

\u3053\u306e\u5f0f\u306f\u6b21\u51433<\/strong>\u306b\u56fa\u6709\u3067\u3059\u3002\u3053\u308c\u306f\u3001\u56de\u8ee2<\/a><\/span>\u5834\u306e\u767a\u6563\u304c\u30bc\u30ed\u3067\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002\u9006\u306b\u3001\u30d9\u30af\u30c8\u30eb\u5834\u306e\u5834\u5408\u3001

$$ {\\vec B} $$<\/div>\u306e\u4e0a
$$ {\\R^3} $$<\/div>\u661f\u304c\u3061\u308a\u3070\u3081\u3089\u308c\u305f\u30aa\u30fc\u30d7\u30f3
$$ {\\R^3} $$<\/div>\u767a\u6563\u304c\u30bc\u30ed\u306e\u5834\u5408\u3001\u30d5\u30a3\u30fc\u30eb\u30c9\u304c\u5b58\u5728\u3057\u307e\u3059\u3002
$$ {\\vec A} $$<\/div>\u306e\u3088\u3046\u306a
$$ {\\vec B=\\vec{\\mathrm{rot}}\\vec A} $$<\/div> \u3002 \uff08\u305d\u306e\u5f8c\u3001\u79c1\u305f\u3061\u306f\u3053\u3046\u8a00\u3044\u307e\u3059
$$ {\\vec A} $$<\/div>\u306f\u30d9\u30af\u30c8\u30eb\u30dd\u30c6\u30f3\u30b7\u30e3\u30eb<\/strong>\u3067\u3059\uff09\u3002\u3053\u306e\u7279\u6027\u306f\u3001\u5fae\u5206\u5f62\u5f0f\u306e\u89b3\u70b9\u304b\u3089\u9069\u5207\u306b\u89e3\u91c8\u3055\u308c\u308b\u3068\u3001\u30dd\u30a2\u30f3\u30ab\u30ec\u306e\u88dc\u984c\u3092\u76f4\u63a5\u9069\u7528\u3067\u304d\u307e\u3059\u3002<\/p>

\u6ce8\u610f\u3002<\/strong>\u30cb\u30e5\u30fc\u30c8\u30f3\u5834

$$ {\\frac{\\vec r}{r^3}} $$<\/div>\u767a\u6563\u306f\u30bc\u30ed\u3067\u3059\u304c\u3001\u30d9\u30af\u30c8\u30eb\u5834\u306f\u3042\u308a\u307e\u305b\u3093
$$ {\\vec A} $$<\/div>\u306e\u3088\u3046\u306a
$$ {\\vec{\\mathrm{rot}}\\vec A=\\frac{\\vec r}{r^3}} $$<\/div> \u3002\u5b9f\u969b\u3001\u3053\u308c\u304c\u5f53\u3066\u306f\u307e\u308b\u5834\u5408\u3001\u9589\u3058\u305f\u8868\u9762\u3092\u901a\u308b\u5149\u675f\u306f\u30bc\u30ed\u306b\u306a\u308a\u307e\u3059\u304c\u3001\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3059\u308b\u7403\u3092\u901a\u308b\u5149\u675f\u306f4\u03c0<\/span>\u306b\u306a\u308a\u307e\u3059\u3002\u5b9f\u969b\u3001\u3053\u306e\u30d5\u30a3\u30fc\u30eb\u30c9\u306f\u3001\u30b9\u30bf\u30fc \u30aa\u30fc\u30d7\u30f3\u3067\u306f\u306a\u3044\u539f\u70b9\u306e\u30d7\u30e9\u30a4\u30d9\u30fc\u30c8\u7a7a\u9593\u3067\u306e\u307f\u5b9a\u7fa9\u3055\u308c\u307e\u3059\u3002\u30dd\u30a2\u30f3\u30ab\u30ec\u306e\u88dc\u984c\u306f\u9069\u7528\u3055\u308c\u307e\u305b\u3093\u3002 <\/p>
$$ {\\mathrm{div}(f\\vec A)=f\\mathrm{div}(\\vec A)+\\nabla f\\cdot \\vec A} $$<\/div><\/dd><\/dl>

\u30b9\u30c8\u30fc\u30af\u30b9\u306e\u5b9a\u7406<\/span>\u306b\u3088\u308c\u3070\u3001\u6b21\u306e\u7a4d\u5206\u306f

$$ {\\R^n} $$<\/div>\u5883\u754c\u90e8\u5206\u306e\u5916\u5074\u306e\u30bc\u30ed\u30d9\u30af\u30c8\u30eb\u5834\u306e\u767a\u6563\u306f\u30bc\u30ed\u3067\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001 f \u304c<\/i><\/span>\u6ed1\u3089\u304b\u306a\u95a2\u6570\u3067\u3042\u308b\u5834\u5408\u3001
$$ {\\vec A} $$<\/div>\u30d9\u30af\u30c8\u30eb\u306e\u30d5\u30a3\u30fc\u30eb\u30c9\u3002\u5883\u754c\u90e8\u5206\u306e\u5916\u5074\u306f\u4e21\u65b9\u3068\u3082\u30bc\u30ed\u3067\u3059 (\u3053\u306e\u6761\u4ef6\u306b\u3088\u308a\u3001\u7a4d\u5206\u304c\u610f\u5473\u3092\u6301\u3064\u3053\u3068\u304c\u4fdd\u8a3c\u3055\u308c\u307e\u3059)\u3002 <\/p>
$$ {\\int_{\\R^n}f\\mathrm{div} \\vec A dx_1\\cdots dx_n=\\int_{\\R^n}\\nabla f\\cdot \\vec A dx_1\\cdots dx_n} $$<\/div><\/dd><\/dl>

\u3053\u306e\u30d7\u30ed\u30d1\u30c6\u30a3\u306f\u6b21\u306e\u3088\u3046\u306b\u89e3\u91c8\u3055\u308c\u307e\u3059\u3002\u3055\u305b\u3066

$$ {C^\\infty(\\R^n)} $$<\/div>\u305d\u3057\u3066
$$ {\\mathrm{Vect(\\R^n)}} $$<\/div>\u305d\u308c\u305e\u308c\u3001\u30b9\u30e0\u30fc\u30ba\u95a2\u6570\u306e\u30d9\u30af\u30c8\u30eb\u7a7a\u9593\u3068\u30d9\u30af\u30c8\u30eb\u5834\u306e
$$ {\\R^n} $$<\/div> \u3002\u79c1\u305f\u3061\u306f\u5f7c\u3089\u306b\u30b9\u30ab\u30e9\u30fc\u88fd\u54c1\u3092\u63d0\u4f9b\u3057\u307e\u3059<\/p>
$$ {\\langle f,g\\rangle\u00a0:=\\int_{\\R^n}fg A dx_1\\cdots dx_n} $$<\/div>\u305d\u3057\u3066
$$ {\\langle \\vec A,\\vec B \\rangle\u00a0:=\\int_{\\R^n}\\vec A\\cdot\\vec B dx_1\\cdots dx_n} $$<\/div><\/dd><\/dl>

\u305d\u308c\u3067

$$ {\\langle \\nabla f,\\vec A\\rangle = -\\langle f,\\mathrm{div}\\vec A\\rangle} $$<\/div>\u3053\u308c\u306b\u3088\u308a\u3001\u767a\u6563\u6f14\u7b97\u5b50\u3092\u52fe\u914d\u6f14\u7b97\u5b50\u306e\u8ee2\u7f6e (\u7b26\u53f7\u307e\u3067) \u3068\u3057\u3066\u898b\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>

\u767a\u6563\u306e\u3053\u306e\u89e3\u91c8\u306b\u306f\u3001\u30ea\u30fc\u30de\u30f3\u591a\u69d8\u4f53\u3068\u30c6\u30f3\u30bd\u30eb\u306e\u4e21\u65b9\u306b\u4e00\u822c\u5316\u3067\u304d\u308b\u3068\u3044\u3046\u5229\u70b9\u304c\u3042\u308a\u307e\u3059\u3002 <\/p>

$$ {\\mathrm{div}(\\vec A\\wedge\\vec B)=\\vec B\\cdot\\mathrm{rot}\\vec A – \\vec A\\cdot\\mathrm{rot}\\vec B} $$<\/div><\/dd><\/dl>

\u3053\u306e\u516c\u5f0f\u306e\u5178\u578b\u7684\u306a\u5fdc\u7528\u4f8b\u306f\u3001\u96fb\u78c1\u6c17\u5b66<\/a><\/span>\u306e\u30dd\u30a4\u30f3\u30c6\u30a3\u30f3\u30b0\u306e\u5b9a\u7406\u3067\u3059\u3002 <\/p>

$$ {\\vec{\\mathrm{rot}}\\left(\\vec{\\mathrm{rot}}\\left(\\vec A\\right)\\right) = \\vec{\\mathrm{grad}}\\left(\\mathrm{div}\\left(\\vec A\\right)\\right)-\\Delta \\left(\\vec{A} \\right)} $$<\/div><\/dd><\/dl>

\u3053\u308c\u3089\u306e\u95a2\u4fc2\u306f\u3001\u30d9\u30af\u30c8\u30eb\u89e3\u6790<\/a><\/span>\u3067\u5e83\u304f\u4f7f\u7528\u3055\u308c\u3066\u304a\u308a\u3001\u5fae\u5206\u5f62\u5f0f\u306e\u30b3\u30f3\u30c6\u30ad\u30b9\u30c8\u3067\u3088\u308a\u3088\u304f\u7406\u89e3\u3055\u308c\u307e\u3059\u3002<\/p>

\u767a\u6563\u306e\u89b3\u70b9\u304b\u3089\u8868\u73fe\u3055\u308c\u305f\u4fdd\u5b58\u5247<\/span><\/h2>